Section 9.1b… Combinations

What are they??? With permutations, we take n objects, r at a time, and different orderings of these objects are considered different permutations. In some situations, different orderings don’t matter!!! Combinations (of n objects taken r at a time) – the numbers of ways to select the objects, regardless of the order in which they are arranged.

Combination Counting Formula The number of combinations of n objects taken r at a time is denoted C and is given by nr for If r > n, then A note about notation: is commonly denoted Both are read: “n choose r ”

Distinguishing Combinations from Permutations In each of the following scenarios, tell whether permutations (ordered) or combinations (unordered) are being described. Then determine the possible choices in the scenario. 1. A president, vice-president, and secretary are chosen from a 25-member garden club. Permutations Permutations – order matters because it matters who gets which office.

Distinguishing Combinations from Permutations In each of the following scenarios, tell whether permutations (ordered) or combinations (unordered) are being described. Then determine the possible choices in the scenario. 2. A cook chooses 5 potatoes from a bag of 12 potatoes to make a potato salad. Combinations Combinations – the salad is the same no matter what order the potatoes are chosen.

Distinguishing Combinations from Permutations In each of the following scenarios, tell whether permutations (ordered) or combinations (unordered) are being described. Then determine the possible choices in the scenario. 3. A teacher makes a seating chart for 22 students in a classroom with 30 desks. Permutations Permutations – a different ordering of students in the same seats results in a different seating chart.

More Guided Practice In the Miss America pageant, 51 contestants must be narrowed down to 10 finalists who will compete on national television. In how many possible ways can the ten finalists be selected? Permutations or Combinations??? ways

More Guided Practice The Georgia Lotto requires winners to pick 6 integers between 1 and 46. The order in which you select them does not matter; indeed, the lottery tickets are always printed with the numbers in ascending order. How many different lottery tickets are possible? Permutations or Combinations??? possible tickets

More Guided Practice Armando’s Pizzeria offers patrons any combination of up to 10 different toppings. How many different pizzas can be ordered possible pizzas (a) if we can choose any three toppings? Order of the toppings does not matter, so:

More Guided Practice Armando’s Pizzeria offers patrons any combination of up to 10 different toppings. How many different pizzas can be ordered (b) if we can choose any number of toppings (0 through 10)? We could add up all numbers of the form for r = 0, 1,…, 10. A quicker way: For each option (topping), we can choose either yes or no. By the Multiplication Principle: possible pizzas

Formula for Counting Subsets of an n - set There are 2 subsets of a set with n objects (including the empty set and the entire set). n

More Guided Practice A national hamburger chain used to advertise that it fixed its hamburgers “256 ways,” since patrons could order whatever toppings they wanted. How many toppings must have been available? We need to solve the following equation for n: toppings